Toward the theory of resolution of singularities in positive characteristic

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This subject is one of famous problems to be solved among modern mathematicians in the world. Most mathematicians may say that it is very difficult to treat this problem. Besides, this grant-in-aid was accepted additionally in the latter half of 1999.Before the acceptance of this grant, I visited Herwig Hauser at Insbruck University to discuss this subject in March 1999. He is one of specialists of this subject in Europe. In July-August 1999 I visited Toronto University and Saskatchewan University in Canada. There I attended at the international symposium of this subject, and discussed with specialists in America and Canada, Eduard Bierstone, Pierre D.Milman, Mark Spivakovsky, Shreeram Abhyankar, and Bernard Teissier from France.Coming back to Japan, I studied very hard to achieve essential advance. I was surprised that tremendous amount of time and effort is necessary to treat this subject. The amount was by far more than that I had expected. I could achieve a better result in the 2dimensional cases than known ones, and wrote the preprint "New ideas for resolution of singularities in arbitrary characteristic". Then, I proceeded to the 3-dimensional cases, the simplest cases for which definite results are not yet known. I spent large amount of time to check large number of items to be checked. I got several essential new ideas.At present, I may be able to reach the final result by my new ideas, and I may encounter serious difficulties on the way to the goal. I do not know which one of these cases occurs. Anyway, I can get some results.To tell the truth, because larger amount of time was necessary than that I had expected, I do not have yet achieved practical results. I would like to continue my efforts to achieve unknown facts.